Let f be continuous on [a,b] and suppose that, for every integrable function g defined on [a,b],∫_{a}^{b}fg=0. Prove that f(x)=0 for all x∈[a,b].

Why isn't g(x)= 0 why does it have to be f(x)? Why not both?

Printable View

- Jul 8th 2011, 10:35 AMCountingPenguinsReimann integral question
Let f be continuous on [a,b] and suppose that, for every integrable function g defined on [a,b],∫_{a}^{b}fg=0. Prove that f(x)=0 for all x∈[a,b].

Why isn't g(x)= 0 why does it have to be f(x)? Why not both? - Jul 8th 2011, 10:53 AMgirdavRe: Reimann integral question
What we suppose works for all Riemann-integrable function , and we only know that .

- Jul 8th 2011, 11:09 AMPlatoRe: Reimann integral question
- Jul 8th 2011, 12:29 PMCountingPenguinsRe: Reimann integral question
f is continuous and g is integrable and the product of the integral of fg over the interval [a,b] is 0. So f(x) must be 0 for all x in [a,b], is all the information I have been given. I will start with the definition of continuous for f and integrable for g and then hope I see something useful after that.

- Jul 8th 2011, 12:45 PMPlatoRe: Reimann integral question
Did you see reply #3. That answers the question.

- Jul 10th 2011, 01:49 PMAlso sprach ZarathustraRe: Reimann integral question
I came up with the following solution, I hope it is not so wrong. (Itwasntme)

Suppose that is not for all , then there exists such that (If the is no such , there must be such that .In that case we will look at ).

f is continuous function therefor exists for which for all .

Now we define -'staircase'-characteristic function of open interval .

Hence, we will get:

.

So, we got a contradiction to the fact that if is continuous on and h is staircase function .

But we also know that if is integrable function on , so for every exists staircase function so that

Combining these two results we will conclude the needed. - Jul 10th 2011, 03:35 PMPlatoRe: Reimann integral question